I think the effects really depend on what kind of religious figure it is. A religious figure like the pope has Petrine authority which extends beyond the mortal world -- the gates of hell shall not prevail against him; what is bound on earth is bound in heaven and what is loosed on earth is loosed in heaven. If the head of a church like the Mormons, Catholics, Copts and other churches that grant a portion of the divinity's power to him dies, the consequences will obviously be somewhat different, especially in RPG terms, than if the simple political head of the church dies. Until this person is replaced properly, the church loses some of its actual power not merely due to political weakness or strife but because there is nobody with the special powers the title confers.
The Gnostics had even more considerable powers granted to their leaders; the Simonians first under Simon then under Menander, had a head of the church who contained all or part of God himself, or perhaps was a hypostasized angel (it's unclear).
So, there is a considerable range of consequences. There could be an interruption in spell granting power or magical effects until the new head comes into power; souls might be trapped in dead bodies, etc. Lots of possible effects depending on how much power the god has delegated to the head of his church.
Obviously, the more power god has given, the more incentive there is both to fight over the job and to find a replacement immediately.
